Assuming Goldbach's conjecture and denoting by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$, consider the sequence $(u_n)_n$ such that $u_0$ is a given composite and $u_{n+1}:=u_n.r_{0}(u_n)$. If the sequence provably converges, we say $u_0$ is a "citizen", otherwise call it a "rebel". I investigated the considered sequences for $u_0$ below 121 and found some 25 rebels : I let the algorithm go on for 30 steps and made sure the last computed $r_{0}$ was not 1. This suggests there are more citizens that rebels, and actually a proof of the existence of infinitely many citizens would imply the twin prime conjecture.

My question is : are there solid arguments leading to a positive density of citizens among the composites? If yes can we estimate a lower bound thereof?


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